Questions: Quiz 15.6.4: Damped oscillation and the mechanical energy An object on a horizontal spring with a spring constant k is initially pulled to the right by a distance A from its equilibrium position. When let go, the object starts oscillating, and its maximum displacement from its equilibrium position is given by A(t) = A e^(-t / τ). Which of the following is the correct expression for the mechanical energy of the object-spring system as a function of time? A. E = 1/2 k(A e^(-t / τ))^2 B. E = 1/2 k A e^(-t / τ) C. E = k A e^(-t / τ) D. E = k(A e^(-t / τ))^2 E. E = 1/2 k A^2

Quiz 15.6.4: Damped oscillation and the mechanical energy An object on a horizontal spring with a spring constant k is initially pulled to the right by a distance A from its equilibrium position. When let go, the object starts oscillating, and its maximum displacement from its equilibrium position is given by A(t) = A e^(-t / τ). Which of the following is the correct expression for the mechanical energy of the object-spring system as a function of time? A. E = 1/2 k(A e^(-t / τ))^2 B. E = 1/2 k A e^(-t / τ) C. E = k A e^(-t / τ) D. E = k(A e^(-t / τ))^2 E. E = 1/2 k A^2

Transcript text: Quiz 15.6.4: Damped oscillation and the mechanical energy An object on a horizontal spring with a spring constant $k$ is initially pulled to the right by a distance $A$ from its equilibrium position. When let go, the object starts oscillating, and its maximum displacement from its equilibrium position is given by $\boldsymbol{A}(\boldsymbol{t})=\boldsymbol{A} \boldsymbol{e}^{-\boldsymbol{t} / \tau}$. Which of the following is the correct expression for the mechanical energy of the object-spring system as a function of time? A. $E=\frac{1}{2} k\left(A e^{-t / \tau}\right)^{2}$ B. $E=\frac{1}{2} k A e^{-t / \tau}$ C. $E=k A e^{-t / \tau}$ D. $E=k\left(A e^{-t / \tau}\right)^{2}$ E. $E=\frac{1}{2} k A^{2}$

Solution

Solution Steps

Step 1: Identify the given information

The spring constant is $ k $.
The initial displacement is $ A $.
The displacement as a function of time is $ A(t) = A e^{-t / \tau} $.

Step 2: Recall the formula for mechanical energy in a spring system

The mechanical energy $ E $ in a spring system is given by the formula: \[ E = \frac{1}{2} k x^2 \] where $ x $ is the displacement from the equilibrium position.

Step 3: Substitute the given displacement function into the energy formula

Substitute $ x = A e^{-t / \tau} $ into the energy formula: \[ E = \frac{1}{2} k (A e^{-t / \tau})^2 \]

Step 4: Simplify the expression

Simplify the expression to get the final form of the mechanical energy as a function of time: \[ E = \frac{1}{2} k A^2 e^{-2t / \tau} \]

Step 5: Match the simplified expression with the given options

Compare the simplified expression with the given options. The correct option is: \[ \text{A. } E = \frac{1}{2} k \left(A e^{-t / \tau}\right)^2 \]